Bright and dark spatial solitons in non-Kerr media

An overview of the theory of self-guided optical beams, spatial optical solitons supported by non-Kerr non-linearities, is presented. This includes bright and dark solitons in optical media with intensity-dependent non-linear response as well as two-component solitary waves supported by parametric wave mixing in quadratic or cubic media. The properties of non-linear spatially localized waves are discussed for qualitatively different types of soliton bearing non-integrable non-linear models, including the scalar model described by a generalized non-linear Schrödinger equation and the models of the second- and third-harmonic generation. Special attention is paid to the recent advances of the theory of soliton stability and soliton internal modes.     

From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.     

Fifth order evolution equation for long wave dissipative solitons

Third and fifth order nonlinear wave equations which arise in the theory of water waves possess solitary and periodic traveling waves. Solitary waves also arise in systems with dissipation and instability where a balance between these effects allows the existence of dissipative solitons. Here we search for a model equation to describe long wave dissipative solitons including fifth order dispersion. The equation found includes quadratic and cubic nonlinearities. For periodic solutions in a small box we characterize the rate of growth, and show that they do not blow up in finite time. Analytic solutions are constructed for special parameter values.     

Generation of acoustic solitons by a single-mode electromagnetic field

Mechanisms of acoustic pulse generation by a single-mode electromagnetic field propagating in a photoelastic material are analyzed. The anisotropy induced by acoustic excitations in an isotropic medium leads to nonlinear coupling between the polarization components of a single-mode electromagnetic field. For different conditions, it is shown that the acoustic-electromagnetic wave interaction due to mixing of the polarization components of light and acoustic waves can give rise to soliton-like coherent acoustic excitations in a thin crystal plate. When spatial dispersion is ignored, the governing system of equations for unidirectional acoustic solitons can be reduced to an integrable model. It is shown that qualitatively different scenarios of formation of acoustic solitons are possible, depending on the directions of deformation and field polarization.     

Bright matter-wave soliton collisions in a harmonic trap: regular and chaotic dynamics

Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.     

Video solitons in a dispersive transmission line with the nonlinear capacitance of a p-n junction

Possible types of low-frequency electromagnetic solitary waves in a dispersive LC transmission line with a quadratic or cubic capacitive nonlinearity are investigated. The fourth-order nonlinear wave equation with ohmic losses is derived from the differential-difference equations of the discrete line in the continuum approximation. For a zero-loss line, this equation can be reduced to the nonlinear equation for a transmission line, the double dispersion equation, the Boussinesq equations, the Korteweg-de Vries (KdV) equation, and the modified KdV equation. Solitary waves in a transmission line with dispersion and dissipation are considered.     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections

We study the propagation of bright two-dimensional spatio-temporal solitary waves using a higher-order multi-dimensional non-linear Schrödinger equation. Starting directly from Maxwell's equations, a multiple-scales derivation is presented which results in a generalized first-order vectorial evolution equation that is valid for the non-linear spatio-temporal propagation of a predominantly linearly polarized electric field with large spatial and temporal bandwidths. A reduced version of this full equation including the higher-order linear and non-linear effects of third- and fourth-order dispersion, space–time focusing, shock, stimulated Raman scattering, and ultrafast quintic index saturation, is solved numerically via a modified split-step algorithm. Material parameters corresponding to those of fused silica at λf=1.55 m are used, with the addition of a negative quintic saturation term. Without quintic saturation, the non-linear spatio-temporal wave broadens under the action of the higher-order space–time effects. In addition, in the absence of Raman scattering, the wave undergoes collapse until arrested by the remaining higher-order terms. Frequency down-shifting and spatio-temporal broadening due to Raman scattering are found to have the greatest effect on non-linear spatio-temporal wave propagation. Nevertheless, we demonstrate that quintic saturation effectively stabilizes the wave such that broadening is reduced considerably, permitting nearly stationary propagation over many confocal distances, albeit with substantial down-shift. The resulting spatio-temporal solitary waves should be useful for applications in ultrafast all-optical switching and logic, and the generalized evolution equations will provide a refined starting point for the study of spatio-temporal phenomena in other areas as well.     

Modification of solitary waves by third-harmonic generation

The effect of phase-matched third-harmonic generation on the structure and stability of spatial solitary waves is investigated. A power threshold for the existence of two-frequency spatial solitons is found, and the multistability of solitary waves in a Kerr medium owing to a higher-order nonlinear phase shift caused by cascaded third-order processes is revealed.     

负折射介质中孤波间相互作用

以描述负折射介质中超短脉冲传输的归一化非线性薛定谔方程为模型,采用对称分步傅里叶算法研究了负折射介质中亮、暗孤波间的相互作用.数值模拟发现:当孤波的初始频移为零时,亮孤波间的相互作用与常规介质中类似;当孤波的初始频移不为零时,其传输速度和相互作用明显受三阶色散和自陡峭效应的影响,主要表现为相互排斥.而负折射介质中暗孤波间的相互作用与常规介质中的相互作用类似,无论暗孤波是否存在初始频移,暗孤波间的相互作用在三阶色散和自陡峭的影响下都表现为相互排斥.结果表明,通过调节三阶色散和自陡峭系数可以在一定程度上抑制负折射介质中亮、暗孤波间的相互作用.该研究结果为负折射介质在未来高速通信中的应用提供了理论依据.     

Stability of solitary wave trains in Hamiltonian wave systems

A class of Hamiltonian nonlinear wave equations possessing complex solitary waves with exponential decay is studied. It is shown that the interpulse interactions in a train of nearly identical solitary waves with large separations between the individual solitary waves are approximately described by a double Toda lattice system, with two variables at each lattice site. Under certain conditions, which are explicitly identified as Cauchy-Riemann equations, the two dynamical variables are real and imaginary parts of a single complex variable, leading to the complex Toda lattice equations, which is a discrete integrable dynamical system. This analysis generalizes to certain nonintegrable partial differential equations a recent result for the nonlinear Schr?dinger equation, and is important for the study of nonlinear communications channels in optical fibers. An example, the cubic-quintic nonlinear Schr?dinger equation, is worked out in detail to show that the theory can be carried through analytically. The theory is used to determine the stability of an infinite chain of nearly identical pulses separated by large time intervals. The entire theory is nonperturbative in the sense that the nonlinear wave equation need not be a weak perturbation of an integrable one.     

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons

An averaged variational principle is applied to analyze the nonlinear effect of transverse perturbations (including diffraction) on quasi-one-dimensional soliton propagation governed by various wave equations. It is shown that parameters of the spatiotemporal solitons described by the cubic Schrödinger equation and the Yajima-Oikawa model of interaction between long-and short-wavelength waves satisfy the spatial quintic nonlinear Schrödinger equation for a complex-valued function composed of the amplitude and eikonal of the soliton. Three-dimensional solutions are found for two-component “bullets” having long-and short-wavelength components. Vortex and hole-vortex structures are found for envelope solitons and for two-component solitons in the regime of resonant long/short-wave coupling. Weakly nonlinear behavior of transverse perturbations of one-dimensional soliton solutions in a self-defocusing medium is described by the Kadomtsev-Petviashvili equation. The corresponding rationally localized “lump” solutions can be considered as secondary solitons propagating along the phase fronts of the primary solitons. This conclusion holds for primary solitons described by a broad class of nonlinear wave equations.     

Twistor correspondences for the soliton hierarchies

In this article we propose a new overview on the theory of integrable systems based on symmetry reduction of the anti-self-dual Yang—Mills equations and its twistor correspondence. First, the non-linear Schrödinger (NS) equations and the Korteweg de Vries (KdV) equations are shown to be symmetry reductions of the anti-self-dual Yang—Mills (ASDYM) equation with real forms of SL (2, ) as gauge groups.

We obtain a twistor correspondence between solutions of the NS and KdV equations and certain holomorphic vector bundles with a symmetry on the total space of the complex line bundle of Chern class two on the Riemann sphere. Remarkably, when the Chern class is increased, the correspondence extends to the NS and KdV hierarchies. If the symmetry condition is dropped we obtain a twistor correspondence for a hierarchy for the Bogomolny equations, which yields the KdV and NS hierarchies when the symmetry is imposed.

The inverse scattering transform is shown to be a coordinate realization of the twistor correspondence. Both the pure solitons and the solitonless cases are treated. The k-soliton solutions arise from the kth “Ward ansatze” in an analogous fashion to the monopole solutions.     

Nonlinear dynamics of the interface between fluids at the suppression of Kelvin–Helmholtz instability by a tangential electric field

The nonlinear dynamics of the interface between ideal dielectric fluids in the presence of tangential discontinuity of the velocity at the interface and the stabilizing action of the horizontal electric field is examined. It is shown that the regime of motion of the interface where liquids move along the field lines occurs in the state of neutral equilibrium where electrostatic forces suppress Kelvin–Helmholtz instability. The equations of motion of the interface describing this regime can be reduced to an arbitrary number of ordinary differential equations describing the propagation and interaction of structurally stable solitary waves, viz. rational solitons. It is shown that weakly interacting solitary waves recover their shape and velocity after collision, whereas strongly interacting solitary waves can form a wave packet (breather).     

Nonlinear Longitudinal Strain Waves in a Solid Exposed to Pulsed Laser Radiation

The propagation of longitudinal strain waves in a solid with quadratic nonlinearity of elastic continuum was studied in the context of a model that takes into account the joint dynamics of elastic displacements in the medium and the concentration of the laser-induced point defects. The input equations of the problem are reformulated in terms of only the total displacements of the medium points. In this case, the presence of structural defects manifests itself in the emergence of a delayed response of the system to the propagation of the strain-related perturbations, which is characteristic of media with relaxation or memory. The model equations describing the nonlinear displacement wave were derived with allowance made for the values of the relaxation parameter. The influence of the generation, relaxation, and the strain-induced drift of defects and the flexoelectricity on the propagation of this wave was analyzed. It is shown that, for short relaxation times of defects, the strain can propagate in the form of both shock fronts and solitary waves (solitons). Exact solutions depending on the type of relation between the coefficients in the equation and describing both the shock-wave structures and the evolution of solitary waves are presented. In the case of longer relaxation times, shock waves do not form and the strain wave propagates only in the form of solitary waves or a train of solitons. The contributions of the finiteness of the defect-recombination rate and the flexoelectricity to linear elastic moduli and spatial dispersion are determined.     

Solitons and spectral renormalization methods in nonlinear optics

Localized wave solutions, often referred to as solitary waves or solitons, are important classes of solutions in nonlinear optics. In optical communications, weakly nonlinear, quasi-monochromatic waves satisfy the “classical” and the “dispersion-managed” nonlocal nonlinear Schrödinger equations, both of which have localized pulses as special solutions. Recent research has shown that mode-locked lasers are also described by similar equations. These systems are variants of the classical nonlinear Schrödinger equation, appropriately modified to include terms which model gain, loss and spectral filtering that are present in the laser cavity. To study their remarkable properties, a computational method is introduced to find localized waves in nonlinear optical systems governed by these equations.     

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg-de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov-Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.     

Completely integrable models of nonlinear optics

The models of the nonlinear optics in which solitons appeared are considered. These models are of paramount importance in studies of nonlinear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency and parametric interaction of three waves. At present there are a number of theories based on completely integrable systems of equations, which are, both, generations of the original known models and new ones. The modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation. Sine-Gordon equation, the reduced Maxwell-Bloch equation. Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are some soliton equations, which are usually to be found in nonlinear optics theory.     

Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications

We present an overview of nonlinear phenomena related to optical quadratic solitons—intrinsically multi-component localized states of light, which can exist in media without inversion symmetry at the molecular level. Starting with presentation of a few derivation schemes of basic equations describing three-wave parametric wave mixing in diffractive and/or dispersive quadratic media, we discuss their continuous wave solutions and modulational instability phenomena, and then move to the classification and stability analysis of the parametric solitary waves. Not limiting ourselves to the simplest spatial and temporal quadratic solitons we also overview results related to the spatio-temporal solitons (light bullets), higher order quadratic solitons, solitons due to competing nonlinearities, dark solitons, gap solitons, cavity solitons and vortices. Special attention is paid to a comprehensive discussion of the recent experimental demonstrations of the parametric solitons including their interactions and switching. We also discuss connections of quadratic solitons with other types of solitons in optics and their interdisciplinary significance.